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A075869 Numbers k such that 5*k^2 - 9 is a square. 0
3, 51, 915, 16419, 294627, 5286867, 94868979, 1702354755, 30547516611, 548152944243, 9836205479763, 176503545691491, 3167227616967075, 56833593559715859, 1019837456457918387, 18300240622682815107 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Lim. n-> Inf. a(n)/a(n-1) = phi^6 = 9 + 4*sqrt(5).
REFERENCES
A. H. Beiler, "The Pellian", ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover, New York, New York, pp. 248-268, 1966.
L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, pp. 341-400.
Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, pp. 139-147.
LINKS
Tanya Khovanova, Recursive Sequences
J. J. O'Connor and E. F. Robertson, Pell's Equation [From the Internet Archive Wayback machine]
Eric Weisstein's World of Mathematics, Pell Equation.
FORMULA
a(n) = 3*sqrt(5)/10*((2+sqrt(5))^(2*n-1)-(2-sqrt(5))^(2*n-1)) = 18*a(n-1) - a(n-2).
G.f.: 3*x*(1-x)/(1-18*x+x^2). [Philippe Deléham, Nov 17 2008; corrected by Georg Fischer, May 15 2019]
MATHEMATICA
LinearRecurrence[{18, -1}, {3, 51}, 20] (* Harvey P. Dale, Dec 27 2018 *)
CROSSREFS
Cf. 3*A007805.
Sequence in context: A248341 A145242 A182512 * A361051 A307369 A126685
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified June 13 10:24 EDT 2024. Contains 373383 sequences. (Running on oeis4.)